= Total branch length of a neutral coalescent
{title2=$L=\sum_{j=2}^n jT_j$}
For $n\geq2$, in coalescent units, <independent> waiting times satisfy $T_j\sim\operatorname{Exp}(j(j-1)/2)$, and total branch length is $L=\sum_{j=2}^n jT_j$. Therefore $EL=2\sum_{i=1}^{n-1}1/i$. The variables $jT_j/2$ are <independent> <exponential distribution>[exponentials] with rates $j-1$, so $W=L/2$ has the same distribution as the maximum of $n-1$ <independent> unit <exponential distribution>[exponentials], with density $(n-1)e^{-w}(1-e^{-w})^{n-2}$. The last identity follows either from its <Laplace transform> or from the <independent> spacings of exponential <order statistic>[order statistics].
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